Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 UsableRulesReductionPairsProof (⇔)
↳4 QDP
↳5 RFCMatchBoundsDPProof (⇔)
↳6 TRUE

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a(f, a(g, a(f, x))) → a(f, a(g, a(g, a(f, x))))
a(g, a(f, a(g, x))) → a(g, a(f, a(f, a(g, x))))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A(f, a(g, a(f, x))) → A(f, a(g, a(g, a(f, x))))
A(f, a(g, a(f, x))) → A(g, a(g, a(f, x)))
A(g, a(f, a(g, x))) → A(g, a(f, a(f, a(g, x))))
A(g, a(f, a(g, x))) → A(f, a(f, a(g, x)))

The TRS R consists of the following rules:

a(f, a(g, a(f, x))) → a(f, a(g, a(g, a(f, x))))
a(g, a(f, a(g, x))) → a(g, a(f, a(f, a(g, x))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) UsableRulesReductionPairsProof (EQUIVALENT transformation)

First, we A-transformed [FROCOS05] the QDP-Problem. Then we obtain the following A-transformed DP problem.
The pairs P are:

f1(g(f(x))) → f1(g(g(f(x))))
f1(g(f(x))) → g1(g(f(x)))
g1(f(g(x))) → g1(f(f(g(x))))
g1(f(g(x))) → f1(f(g(x)))

and the Q and R are:
Q restricted rewrite system:
The TRS R consists of the following rules:

g(f(g(x))) → g(f(f(g(x))))
f(g(f(x))) → f(g(g(f(x))))

Q is empty.

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(f(x1)) = x1   
POL(f1(x1)) = 2·x1   
POL(g(x1)) = x1   
POL(g1(x1)) = 2·x1   

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

f1(g(f(x))) → f1(g(g(f(x))))
f1(g(f(x))) → g1(g(f(x)))
g1(f(g(x))) → g1(f(f(g(x))))
g1(f(g(x))) → f1(f(g(x)))

The TRS R consists of the following rules:

g(f(g(x))) → g(f(f(g(x))))
f(g(f(x))) → f(g(g(f(x))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(5) RFCMatchBoundsDPProof (EQUIVALENT transformation)

Finiteness of the DP problem can be shown by a matchbound of 2.
As the DP problem is minimal we only have to initialize the certificate graph by the rules of P:

f1(g(f(x))) → f1(g(g(f(x))))
f1(g(f(x))) → g1(g(f(x)))
g1(f(g(x))) → g1(f(f(g(x))))
g1(f(g(x))) → f1(f(g(x)))

To find matches we regarded all rules of R and P:

g(f(g(x))) → g(f(f(g(x))))
f(g(f(x))) → f(g(g(f(x))))
f1(g(f(x))) → f1(g(g(f(x))))
f1(g(f(x))) → g1(g(f(x)))
g1(f(g(x))) → g1(f(f(g(x))))
g1(f(g(x))) → f1(f(g(x)))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1189894, 1189895, 1189897, 1189896, 1189898, 1189899, 1189900, 1189901, 1189902, 1189903, 1189904, 1189905, 1189908, 1189906, 1189907, 1189910, 1189911, 1189909, 1189912, 1189913, 1189914, 1189915, 1189916, 1189917

Node 1189894 is start node and node 1189895 is final node.

Those nodes are connect through the following edges:

  • 1189894 to 1189896 labelled g1_1(0)
  • 1189894 to 1189898 labelled f1_1(0)
  • 1189894 to 1189901 labelled f1_1(0)
  • 1189894 to 1189903 labelled g1_1(0)
  • 1189895 to 1189895 labelled #_1(0)
  • 1189897 to 1189895 labelled f_1(0)
  • 1189897 to 1189906 labelled f_1(1)
  • 1189896 to 1189897 labelled g_1(0)
  • 1189896 to 1189909 labelled g_1(1)
  • 1189898 to 1189899 labelled g_1(0)
  • 1189899 to 1189900 labelled g_1(0)
  • 1189899 to 1189909 labelled g_1(1)
  • 1189900 to 1189895 labelled f_1(0)
  • 1189900 to 1189906 labelled f_1(1)
  • 1189901 to 1189902 labelled f_1(0)
  • 1189901 to 1189906 labelled f_1(1)
  • 1189902 to 1189895 labelled g_1(0)
  • 1189902 to 1189909 labelled g_1(1)
  • 1189903 to 1189904 labelled f_1(0)
  • 1189904 to 1189905 labelled f_1(0)
  • 1189904 to 1189906 labelled f_1(1)
  • 1189905 to 1189895 labelled g_1(0)
  • 1189905 to 1189909 labelled g_1(1)
  • 1189908 to 1189895 labelled f_1(1)
  • 1189908 to 1189906 labelled f_1(1)
  • 1189906 to 1189907 labelled g_1(1)
  • 1189907 to 1189908 labelled g_1(1)
  • 1189907 to 1189909 labelled g_1(1)
  • 1189907 to 1189912 labelled g_1(2)
  • 1189910 to 1189911 labelled f_1(1)
  • 1189910 to 1189906 labelled f_1(1)
  • 1189910 to 1189915 labelled f_1(2)
  • 1189911 to 1189895 labelled g_1(1)
  • 1189911 to 1189909 labelled g_1(1)
  • 1189909 to 1189910 labelled f_1(1)
  • 1189912 to 1189913 labelled f_1(2)
  • 1189913 to 1189914 labelled f_1(2)
  • 1189914 to 1189907 labelled g_1(2)
  • 1189915 to 1189916 labelled g_1(2)
  • 1189916 to 1189917 labelled g_1(2)
  • 1189917 to 1189910 labelled f_1(2)

(6) TRUE